Chapter 7.4, Problem 24E

To determine

The numbers of bacteria were present initially

Answer to Problem 24E

The numbers of bacteria were present initially $=1250$

Explanation of Solution

Given information: A colony of bacteria is grown under ideal conditions in a laboratory so that the population increases exponentially with time. At the end of $\text{3h}$ there are $\text{10,000}$ bacteria.At the end of $\text{5h}$ there are $\text{40,000}$ bacteria

Formula used:

  $y=y_0{e}^{-kt}$

  $y_0=$ Number of bacteria present initially

  $y=$ Number of bacteria present after $t$ time

Where $k$ is a constant

Calculation:

At the end of $\text{3h}$ there are $\text{10,000}$ bacteria

  $y=y_0{e}^{-kt}$

  $y_0=$ Number of bacteria present initially

  $y=$ Number of bacteria present after $t$ time

Here, $\text{t=3,y=10,000}$

Then,

  $10000=y_0{e}^{-3k}$

  $\text{equation1}$

At the end of $\text{5h}$ there are $\text{40,000}$ bacteria

Here, $\text{t=5,y=40,000}$

Then,

  $40000=y_0{e}^{-5k}$

  $\text{equation2}$

By dividing $\text{equation1}$ into $\text{equation2}$

  $\begin{array}{l}4=e^{-2k}\\ \ln 4=-2k\\ k=\frac{-1}{2}\ln 4\\ k=-\ln 2\end{array}$

Putting the value of $k$ in $\text{equation1}$

  $\begin{array}{l}10000=y_0{e}^{3\ln 2}\\ y_0{e}^{3\ln 2}=10000\\ y_0=10000e^{-3\ln 2}\\ y_0=1250\end{array}$

The numbers of bacteria were present initially $=1250$